Dark Energy and the Inflection Points of Cosmic
Expansion in Standard and Brane Cosmologies
Daniel Schmidt, Liberty University
Cyclotron Institute--Texas A&M University
Mentor: Dr. Akram Zhanov
Friedman Equations in Standard Cosmology
Upper and lower limits on t:
R ''
4 G
3p 

(  2 ) 
R
3
c
3
tmin = 12 Ga
λ = 0.59
35
tmax = 17 Ga
λ = 0.89
30
Currently accepted value
(from WMAP data)
8 G
 kc
 R'
  2
  
3
3 R
R
2
2
tWMAP = 13.7 Ga
λ = 0.75
25
20
10
5
1
Here Λ is the cosmological constant, or vacuum energy
density, k is the curvature of space, p is pressure, and ρ is
energy density.
The second equation is separable, so we can solve for t as
a function of Λ.
t Ga
40
35
30
25
20
15
10
5
0.5
Observational Evidence for Nonzero λ
15
These equations describe the evolution of the scale factor
R(t), which represents the relative “size” of the universe.
1
t Ga
40
0.5
1
Here the measured values of λ are superimposed on the t
vs. λ graph.
Inflection Point(s) in Brane Cosmology
Superstring theories introduce some corrections to the
Friedman equations, leading to possible changes in the
evolution of the scale factor R. The second equation
below is the corrected Freidman equation according to
the RSII model.
8 G

 R'

  
3
3
R
2
8 G


 R'
 (1  ) 
  
3
b
3
R
2
Here ρb is a constant introduced by the model.
0.5
0.5
1
t vs. λ graph
Note: λ = Λ 8 π G ρc is a dimensionless “reduced”
cosmological constant, where ρc is the critical density.
(1 Ga = 1 billion years)
The History of Cosmic Expansion
Returning to the first Friedman equation, we see that at
earlier times in the history of the universe, the matter
and radiation density would have been more important
than λ, leading to deceleration.
In the present universe, the expansion is known to be
accelerating.
Somewhere in the middle, there must exist an
inflection point at which the acceleration is
momentarily zero. (See graph at right)
This can be found by setting R’’ = 0, and solving for R.
Note: k has been set to zero since observations indicate
that the universe is flat.
Multiply this by R8, differentiate, and set R’’ = 0.
2 0 r om
8 7 4
x x '- G0 m (

x  x 4 ) x '3
3
b
b
0 m
4
G (  0 r   0 m x)(
x ' 4 x 3 x ')  0
3
b
Here x = R/R0. The resulting seventh order polynomial
equation has one real solution, as in the standard
cosmology. Thus, it seems that the results found in the
standard cosmology are valid in brane cosmology as well.
Without λ, the universe would be younger than
some of its oldest stars.
Other evidence for λ comes from observations
of high redshift supernovae, and gravitational
lensing statistics.
Type Ia supernovae are used as “standard
candles” to measure distance independently of
redshift. This allows the redshift-distance
relation to be measured, which in turn yields
the history of cosmic expansion.
Gravitational lensing of high redshift objects is
another test. The number of condensed,
lensing bodies per unit volume depends on the
scaling factor. Measuring the evolution of this
number allows one to observe the effects of λ
on the expansion.
Measured values are summarized at right.
Paper
Technique
Value
Limit
Kochanek 1996
Gravitational lensing
statistics
NA
<0.66 (95%
confidence)
Myungshin et al. 1997
Gravitational lensing
statistics
0.64 (+0.15, -0.26)
NA
Chiba & Yoshii 1997
Gravitational lensing
statistics
0.8
NA
Chiba & Yoshii 1999
Gravitational lensing
statistics
0.7 (+0.1, -0.2)
NA
Perlmutter et al. 1997
High-Z supernovae
0.06 (+0.28, -0.34)
<0.51 (95%
confidence)
Riess et al. 1998
High-Z supernovae
0.68 (+-0.10)
0.84 (+-0.09)
>0 (98% confidence)
Perlmutter et al. 1999
High-Z supernovae
0.71 (+0.08, -0.09)
>0 (99% confidence)
Inflection Point in Standard Cosmology
R R0
R/R0 vs. t
2.5
2
1.5
1
0.5
0.5
1
1.5
2
t
R ''
4 G
3p


(  2 ) 
R
3
c
3
The precise location of this inflection point cannot
be determined, since the parameter ρb is not known.
It depends on the five dimensional Plank mass as
follows:
ρb = 96 π G M56
M5 is not known, but it may be constrained by this
model.
The single inflection point occurs for x < 1 only if
M5 > 1.9 x 1018 eV/c2.
Thus, if the RSII model is correct, we have a
constraint on M5
R ''
4 G
3p 

(  2 ) 
R
3
c
3
3
4
0 r R0
R ''
4 G 0 m R0


(
2
)
3
4
R
3
R
R
3
4 G
8 G
 4
3
4
0  (
0 m R0 ) R  (
0 r R0 )  R
3
3
3
This quartic equation has only one positive real solution:
R = 0.60 R0 corresponding to a time of about 7.2 Ga.
(R0 is the current value of the scale factor.)
This research was supported by NSF grant
PHY463291-00001, the Texas A&M Cyclotron
Institute, and the Department of Energy.
Special thanks to Dr. Akram Zhanov and the Cyclotron
Institute staff.